Suppose that $X_1, ldots, X_n$ are mutlivariate normal with equal correlation

$rho$ and each of them are marginally

distributed as $N(0,1)$. Let $X_{(1)}, ldots, X_{(n)}$

be the corresponding order statistics. The distribution of

$X_{(1)}$ and $X_{(n)}$ are easily found. What about the

distribution of the other order statistics? Can anyone

give some reference on this?

Thank you. Hanna

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#### Best Answer

As shown in **Tong, Y. L. (1990). Multivariate normal distribution. Springer-Verlag., ch. 6**, for the setup described in the question and for *non-negative* correlation coefficient $rhoin [0,;1)$, the distribution function (cdf) and the density of an order statistic $X_{(i)}$ are (where $phi()$ and $Phi()$ are the standard normal pdf and cdf)

$$G_{(i)}(x) = int_{-infty}^{infty}F_{(i)}left(frac{x+sqrt{rho}z}{sqrt{1-rho}}right)phi(z)dz$$

and differentiating,

$$g_{(i)}(x) = int_{-infty}^{infty}frac 1{sqrt{1-rho}}f_{(i)}left(frac{x+sqrt{rho}z}{sqrt{1-rho}}right)phi(z)dz$$

where

$$f_{(i)}(y) = frac{n!}{(i-1)!(n-i)!}[Phi(y)]^{i-1}[Phi(-y)]^{n-i}phi(y)$$ and $$F_{(i)}(y) = sum_{j=i}^n {n choose j}[Phi(y)]^{j}[Phi(-y)]^{n-j}$$

i.e $f_{(i)}(y)$ and $F_{(i)}(y)$ are the pdf and cdf of the order statistic $(i)$ from an i.i.d. standard normal random sample.

For the corresponding results when the correlation coefficient is negative, the author refers to the book **"Order Statistics", by H.A. David & H.N. Nagaraja ch. 5** (now in its 3d edition, 2003).